Nonlinear Sideband Thermocapillary Instability of a Thin Film Coating the Inside of a Thick Walled Cylinder with Finite Thermal Conductivity in the Absence of Gravity

  • L. A. Dávalos-OrozcoEmail author
Original Article


The nonlinear sideband thermocapillary instability of a thin liquid film coating the inside of a heated cylinder in the absence of gravity is investigated. It is shown that for a newtonian fluid and under the approximation of small wavenumber and large radius of the cylinder, the axial and all azimuthal modes with wavenumber kmax > 0 have the same linear maximum growth rate, in the same way as in a previous papers for flow outside the cylinder. Here, this indeterminacy of the linear problem is resolved nonlinearly looking for the parameters’ range where the axial mode prevails and where it is unstable against the first azimuthal mode of thermocapillary instability.


Cylindrical thin liquid film Marangoni convection Thermocapillary convection Nonlinear sideband instability Thick wall Wall finite thermal conductivity 



The author would like to thank Alberto López, Alejandro Pompa, Cain González, Raúl Reyes, Ma. Teresa Vázquez and Oralia Jiménez for technical support.


  1. Assaf, A, Alkharashi, S.A.: Hydromagnetic instability of a thin viscoelastic layer on a moving column. Phys. Scr. 94(045201), 1–19 (2019). CrossRefGoogle Scholar
  2. Benjamin, T.B., Hasselmann, K.: Instability of periodic wavetrains in nonlinear dispersive systems [and Discusssion]. Proc. Royal Soc. London A 299-1456, 59–76 (1967). Google Scholar
  3. Benney, D.J.: Long waves on liquid films. J. Math. Phys. 45, 150–155 (1966). MathSciNetCrossRefzbMATHGoogle Scholar
  4. Binnie, A.M.: Experiments on the onset of wave formation on a film of water flowing down a vertical plane. J. Fluid Mech. 2, 551–553 (1957). CrossRefGoogle Scholar
  5. Bocharov, A.A., Tsveloduv, O.Yu.: Wave regimes of viscous film flow down a vertical cylinder. Fluid Dyn. 38, 321–327 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chao, Y.C., Ding, Z.Y., Liu, R.: Dynamics of thin liquid films flowing down the uniformly heated/cooled cylinder with wall slippage. Chem. Eng. Sci. 175, 354–364 (2018). CrossRefGoogle Scholar
  7. Cheng, M.Q., Chang, H.-C.: Stability of axisymmetric waves on liquid films flowing down a vertical column to azimuthal and streamwise disturbances. Chem. Eng. Comm. 118, 327–340 (1992). CrossRefGoogle Scholar
  8. Dávalos-Orozco, L.A.: The effect of the thermal conductivity and thickness of the wall on the nonlinear instability of a thin film flowing down an incline. Int. J. Nonlinear Mech. 47, 1–7 (2012). CrossRefGoogle Scholar
  9. Dávalos-Orozco, L.A.: Nonlinear instability of a thin film flowing down a smoothly deformed thick wall of finite thermal conductivity. Interfacial Phenomena and Heat Transfer 2, 55–74 (2014). CrossRefGoogle Scholar
  10. Dávalos-Orozco, L.A.: Non-linear instability of a thin film flowing down a cooled wavy thick wall of finite thermal conductivity. Phys. Lett. 379, 962–967 (2015). CrossRefGoogle Scholar
  11. Dávalos-Orozco, L.A.: Thermal Marangoni instability of a thin film flowing down a thick wall deformed in the backside. Phys. Fluids 28(054103), 1–12 (2016). CrossRefGoogle Scholar
  12. Dávalos-Orozco, L. A.: Sideband thermocapillary instability of a thin film coating the outside of a thick walled cylinder with finite thermal conductivity in the absence of gravity. Interfacial Phenomena and Heat Transfer 5, 287–298 (2017). CrossRefGoogle Scholar
  13. Dávalos-Orozco, L.A.: Sideband thermocapillary instability of a thin film flowing down the inside of a thick walled cylinder with finite thermal conductivity. Interfacial Phenomena and Heat Transfer 6, 239–251 (2018). CrossRefGoogle Scholar
  14. Dávalos-Orozco, L.A.: Sideband thermocapillary instability of a thin film flowing down the outside of a thick walled cylinder with finite thermal conductivity. Int. J. Non-Linear Mech. 109, 15–23 (2019). CrossRefGoogle Scholar
  15. Dávalos-Orozco, L. A., Ruiz-Chavarría, G.: Hydrodynamic instability of a fluid layer flowing down a rotating cylinder. Phys. Fluids A 5, 2390–2404 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dávalos-Orozco, L.A., You, X.: Three-dimensional instability of a liquid layer flowing down a heated vertical cylinder. Physics Fluids 12, 2198–2209 (2000). CrossRefzbMATHGoogle Scholar
  17. Dijkstra, H.A.: The coupling of marangoni and capillary instabilities in an annular thread of liquid. J. Colloid Interface Sci. 136, 151–159 (1990). CrossRefGoogle Scholar
  18. Ding, Z.Y., Liu, Z., Liu, R., Yang, C.: Breakup of ultra-thin liquid films on vertical fiber enhanced by Marangoni effect. Chem. Eng. Sci. 199, 342–348 (2019). CrossRefGoogle Scholar
  19. Duan, L.-S., Duan, L., Jiang, H., Kang, Q.: Oscillation transition routes of Buoyant-thermocapillary convection in annular liquid layers. Microgravity Sci. Technol 30, 565–876 (2018). CrossRefGoogle Scholar
  20. Frenkel, A.L., Babchin, A.J., Levich, B.G., Shlang, T., Sivashinsky, G.I.: Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225–233 (1987). CrossRefGoogle Scholar
  21. Frenkel, A.L.: On evolution equations for thin films flowing down solid surfaces. Phys. Fluids A 5, 2342–2347 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ghosh, A., Bandyopadhyay, D., Sharma, A.: Micro-patterning of coatings on a fiber surface exploiting the contact instabilities of thin viscoelastic films. Phys. Fluids 30(114101), 1–18 (2018). CrossRefGoogle Scholar
  23. Hernández-Hernández, I.J., Dávalos-Orozco, L. A.: Competition between stationary and oscillatory viscoelastic thermocapillary convection of a film coating a thick wall. Int. J. Thermal Sci. 89, 164–173 (2015). CrossRefGoogle Scholar
  24. Kolegov, K.S.: Simulation of patterned glass film formation in the evaporating colloidal liquid under IR heating. Microgravity Sci. Technol 30, 113–120 (2018). CrossRefGoogle Scholar
  25. Krishnamoorthy, S., Ramaswamy, B., Joo, S.W.: Spontaneous rupture of thin liquid films due to thermocapillarity: A full-scale direct numerical simulation. Phys. Fluids 7(9), 2291–2293 (1992). CrossRefzbMATHGoogle Scholar
  26. Liu, R., Ding, Z.J., Zhu, Z.Q.: Thermocapillary effect on the absolute and convective instabilities of film flows down a fibre. Int. J. Heat Mass Transfer 112, 918–925 (2017). CrossRefGoogle Scholar
  27. Liu, R., Ding, Z.J., Chen, X.: The effect of thermocapillarity on the dynamics of an exterior coating film flow down a fibre subject to an axial temperature gradient. Int. J. Heat Mass Transfer 123, 718–727 (2018). CrossRefGoogle Scholar
  28. Mo, D.-M., Ruan D.-F.: Linear-stability analysis of thermocapillary-buoyancy convection in an annular two-layer system with upper rigid wall subjected to a radial temperature gradient. Microgravity Sci. Technol. (2019)CrossRefGoogle Scholar
  29. Moctezuma-Sánchez, M., Dávalos-Orozco, L. A.: Linear three dimensional instability of viscoelastic fluid layers flowing down cylindrical walls. Microgravity Sci. Technol 20, 161–164 (2008). CrossRefGoogle Scholar
  30. Moctezuma-Sánchez, M., Dávalos-Orozco, L.A.: Azimuthal instability modes in a viscoelastic liquid layer flowing down a heated cylinder. Int. J. Heat Mass Transfer 90, 15–25 (2015). CrossRefGoogle Scholar
  31. Rietz, M., Scheid, B., Gallaire, F., Kofman, N., Kneer, R., Rohlfs, W.: Dynamics of falling films on the outside of a vertical rotating cylinder: Waves, rivulets and dripping transitions. J. Fluid Mech. 832, 189–211 (2017). CrossRefGoogle Scholar
  32. Ruiz-Chavarría, G., Dávalos-Orozco, L.A.: Stability of a liquid film flowing down a rotating cylinder subject to azimuthal disturbances. J. Phys. II (France) 6, 1219–1227 (1996). CrossRefGoogle Scholar
  33. Ruiz-Chavarría, G., Dávalos-Orozco, L.A.: Azimuthal and streamwise disturbances in a fluid layer flowing down a rotating cylinder. Phys. Fluids 9, 2899–2908 (1997). CrossRefGoogle Scholar
  34. Shlang, T., Sivashinsky, G.I.: Irregular flow of a liquid film down a vertical column. J. Physique 43, 459–466 (1982). CrossRefGoogle Scholar
  35. Tsveloduv, O. Yu., Bocharov, A.A.: Nonstationary periodic spatial waves on the surface of a viscous liquid film falling down a vertical cylinder. Thermophys. Aeromech. 19, 293–306 (2012). CrossRefGoogle Scholar
  36. Tuckerman, L.S., Barkley, D.: Bifurcation analysis of the Eckhaus instability. Physica D 46, 57–86 (1990). MathSciNetCrossRefGoogle Scholar
  37. Yu, Z.: Thermocapillary instability of self-rewetting films on vertical fibers. Phys. Fluids 30(082104), 1–10 (2018). Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de Investigaciones en Materiales, Departamento de PolímerosUniversidad Nacional Autónoma de MéxicoMéxicoMexico

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